Fourier transform technology, for example FFT (Fast Fourier Transform), is widely used in test instruments such as network analyzers and spectrum analyzers as a means for analyzing the response of a device, the frequency spectrum of an incoming signal and the like. For example, such a Fourier transform process is performed on time domain data that is obtained by measuring an incoming signal in a predetermined time interval. Such test instruments convert the time domain data to frequency domain data and analyze the frequency components in the frequency domain or obtain the frequency spectrum of the incoming signal.
Alternatively, frequency domain data may be converted to time domain data through a Fourier transform process (or an inverse Fourier transform process). For example, in measuring a communication device such as a filter or other device under test (DUT), a network analyzer provides a frequency swept signal to the DUT and measures a resultant frequency domain signal in a predetermined frequency step. Based on the measured data, the network analyzer calculates and displays various parameters including transfer functions, reflection coefficients, phase shifts, group delay, Smith chart, etc., of the DUT.
The network analyzer may further be used to obtain a time domain response, such as a time domain reflectometry (TDR), of the DUT. In such a situation, for example, the frequency domain data indicating the transfer function of the DUT may be inverse Fourier transformed into time domain data. Prior to the inverse transform, a window function may be provided to the transfer function in the frequency domain. Consequently, it is possible to analyze a time domain response of the DUT, such as an impulse response without actually applying an impulse to the DUT.
Generally, such a Fourier transform method is based on what is called a discrete Fourier transform where a DUT response is measured in terms of discrete harmonics determined by a sequence of equally spaced samples. A discrete Fourier transform generally requires a large number of calculations. In particular, for N measured data points, N.sup.2 transform coefficients are calculated. Consequently, for large data sets, the discrete transform process may take a long time to complete the calculation.
To address this issue, a high speed Fourier transform method, the so-called Fast Fourier Transform or FFT, was developed by J. W. Cooly and J. W. Tukey. FFT is an algorithm, typically implemented on a computer, used to reduce the number of calculations required to obtain a DFT. In essence, an FFT algorithm reduces the number of calculations of a typical DFT by eliminating redundant operations when dealing with Fourier series. As a result, according to the FFT, the number of operations required is represented by N log.sub.2 N where N is the number of data samples to be transformed. Thus, the FFT requires significantly fewer calculations than that required in the DFT, and for large data arrays, the FFT is considerably faster than the conventional DFT.
There are some drawbacks to FFT methods. First is that the FFT requires the number N of a transform array to be equal to a power of 2, which may prove restrictive for some applications. More importantly however, in general it is very difficult to initiate the FFT transform until all of the N sampled data are taken. Thus, an overall time required for the Fourier transform operation is represented by T.sub.MES +T.sub.FFT as shown in FIG. 8, where T.sub.MES is a measuring time for obtaining all of the sampled data and T.sub.FFT is a Fourier transformation time by the FFT algorithm.
There is another type of Fourier transform process called Chirp Z transform, which is an improved version of FFT, that can perform Fourier transform with higher resolution than that of FFT. Another advantage of the Chirp Z transform is that the number of data samples need not be equal to a power of 2. This Fourier transform method is described by Rabiner and Gold in "Theory and Application of Digital Signal Processing", pages 393-398, 1975. As far as a transformation time is concerned, since the Chirp Z transform process typically carries out the FFT process three times, a transformation time of 3T.sub.FFT must be added to the measuring time T.sub.MES as shown in FIG. 9. In other words, the Chirp Z transform requires a longer Fourier transformation time than that required in the traditional FFT process.